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Triangular number

From AoPSWiki

The triangular numbers are the numbers $\displaystyle T_n$ which are the sum of the first $\displaystyle n$ natural numbers from $\displaystyle 1$ to $\displaystyle n$ .

Using the sum of an arithmetic series formula, a formula can be calculated for $\displaystyle T_n$ :

$T_n = \displaystyle\sum_{k=1}^{n}k = 1 + 2 + \ldots + n = \frac{n(n+1)}2$

The rather simple recursive definition can be easily found by noting that $\displaystyle T_{n} = 1 + 2 + \ldots + (n-1) + n = (1 + 2 + \ldots + n-1) + n = T_{n-1} + n$ .

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